↑

Properties

Label 2-930-155.39-c1-0-3
Degree $2$
Conductor $930$
Sign $0.0525 - 0.998i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 + 0.587i)4-s − 2.23·5-s − 0.999·6-s + (−0.726 + i)7-s + (−0.587 − 0.809i)8-s + (0.809 − 0.587i)9-s + (2.12 + 0.690i)10-s + (1.92 + 1.40i)11-s + (0.951 + 0.309i)12-s + (−0.138 + 0.0450i)13-s + (1 − 0.726i)14-s + (−2.12 + 0.690i)15-s + (0.309 + 0.951i)16-s + (−1.08 − 1.5i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.549 − 0.178i)3-s + (0.404 + 0.293i)4-s − 0.999·5-s − 0.408·6-s + (−0.274 + 0.377i)7-s + (−0.207 − 0.286i)8-s + (0.269 − 0.195i)9-s + (0.672 + 0.218i)10-s + (0.581 + 0.422i)11-s + (0.274 + 0.0892i)12-s + (−0.0384 + 0.0125i)13-s + (0.267 − 0.194i)14-s + (−0.549 + 0.178i)15-s + (0.0772 + 0.237i)16-s + (−0.264 − 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0525 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0525 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0525 - 0.998i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.0525 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534161 + 0.506802i\)
\(L(\frac12)\) \(\approx\) \(0.534161 + 0.506802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + 2.23T \)
31 \( 1 + (2.19 - 5.11i)T \)
good7 \( 1 + (0.726 - i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.92 - 1.40i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.138 - 0.0450i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.08 + 1.5i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.38 - 4.25i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.44 + 4.73i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.11 - 6.51i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (2.38 - 7.33i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.80 - 1.23i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.93 - 0.954i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.53 + 6.23i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.972 - 2.99i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 2.76T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + (-3.23 + 2.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.73 + 2.38i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.92 + 3.57i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.62 + 0.854i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.85 - 1.34i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.555 + 0.763i)T + (-29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15554630313362662982839458632, −9.328360850936659872396809548585, −8.526015989366627662848348540692, −8.004089990531486568821638177870, −7.02496168125096623170745664535, −6.37165898323396694379267928418, −4.77147700683066340496865186143, −3.73508892047316508612923914146, −2.82959794689358377945372871596, −1.47320422078745079868641460257, 0.41329827043486249544429323964, 2.18193084488975344559714906387, 3.59890339842053591187351694823, 4.19643222037304653192584655764, 5.65256835733481980690506468997, 6.76498642185410889666463018560, 7.50745311157730221938456336001, 8.173027287246235022972699703070, 9.045969807496352198704981036171, 9.599164397863877359045389292264

Graph of the $Z$-function along the critical line